import numpy as np
from pylops import LinearOperator
[docs]class CausalIntegration(LinearOperator):
r"""Causal integration.
Apply causal integration to a multi-dimensional array along ``dir`` axis.
Parameters
----------
N : :obj:`int`
Number of samples in model.
dims : :obj:`list`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which smoothing is applied.
sampling : :obj:`float`, optional
Sampling step ``dx``.
halfcurrent : :obj:`float`, optional
Add half of current value (``True``) or the entire value (``False``)
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly (``True``)
or not (``False``)
Notes
-----
The CausalIntegration operator applies a causal integration to any chosen
direction of a multi-dimensional array.
For simplicity, given a one dimensional array, the causal integration is:
.. math::
y(t) = \int x(t) dt
which can be discretised as :
.. math::
y[i] = \sum_{j=0}^i x[j] dt
or
.. math::
y[i] = (\sum_{j=0}^{i-1} x[j] + 0.5x[i]) dt
where :math:`dt` is the ``sampling`` interval. In our implementation, the
choice to add :math:`x[i]` or just :math:`0.5x[i]` is made by selecting
the ``halfcurrent`` parameter.
Note that the integral of a signal has no unique solution, as any constant
:math:`c` can be added to :math:`y`, for example if :math:`x(t)=t^2` the
resulting integration is:
.. math::
y(t) = \int t^2 dt = \frac{t^3}{3} + c
If we apply a first derivative to :math:`y` we in fact obtain:
.. math::
x(t) = \frac{dy}{dt} = t^2
no matter the choice of :math:`c`.
"""
def __init__(self, N, dims=None, dir=-1, sampling=1,
halfcurrent=True, dtype='float64'):
self.N = N
self.dir = dir
self.sampling = sampling
self.halfcurrent = halfcurrent
if dims is None:
self.dims = [self.N, 1]
self.reshape = False
else:
if np.prod(dims) != self.N:
raise ValueError('product of dims must equal N!')
else:
self.dims = dims
self.reshape = True
self.shape = (self.N, self.N)
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
if self.reshape:
x = np.reshape(x, self.dims)
if self.dir != -1:
x = np.swapaxes(x, self.dir, -1)
y = self.sampling * np.cumsum(x, axis=-1)
if self.halfcurrent:
y -= self.sampling * x / 2.
if self.dir != -1:
y = np.swapaxes(y, -1, self.dir)
return y.ravel()
def _rmatvec(self, x):
if self.reshape:
x = np.reshape(x, self.dims)
if self.dir != -1:
x = np.swapaxes(x, self.dir, -1)
xflip = np.flip(x, axis=-1)
if self.halfcurrent:
y = self.sampling * (np.cumsum(xflip, axis=-1) - xflip/2.)
else:
y = self.sampling * np.cumsum(xflip, axis=-1)
y = np.flip(y, axis=-1)
if self.dir != -1:
y = np.swapaxes(y, -1, self.dir)
return y.ravel()